Abstract

We present a class forcing notion, uniformly definable for ordinals η, which forces the ground model to be the ηth inner mantle of the extension, in which the sequence of inner mantles has length at least η. This answers a conjecture of Fuchs, Hamkins, and Reitz [1] in the positive. We also show that forces the ground model to be the ηth iterated of the extension, where the sequence of iterated s has length at least η. We conclude by showing that the lengths of the sequences of inner mantles and of iterated s can be separated to be any two ordinals you please.